Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions

TL;DR

This paper proves a rigidity theorem for scalar curvature on odd-dimensional singular manifolds with cone-like singularities, extending previous even-dimensional results using advanced analysis and spectral flow techniques.

Contribution

It introduces a new approach applying spectral flow and analysis of cone operators to establish scalar curvature rigidity in odd dimensions.

Findings

Established a Llarull-type rigidity for scalar curvature on singular manifolds in odd dimensions.

Extended analysis of cone operators to twisted Dirac operators on singular spaces.

Combined spectral flow with cone operator analysis to prove the main rigidity result.

Abstract

The main result of this article is a Llarull-type rigidity statement for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions. The even dimensional analog was proven in an earlier work together with Simone Cecchini, Bernhard Hanke and Thomas Schick using index theory and the analysis of abstract cone operators, which applies to Dirac operators associated with generalized cone metrics. We will extend the analysis of abstract cone operators, apply it to twisted Dirac operators on singular manifolds and combine it with a spectral flow argument to prove the main result.